P. 148 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14701-14800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cntoptop 14701	The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- J e. Top
Theorem	cnbl0 14702	Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
|- D = ( abs o. - )   =>    |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` D ) R ) )
Theorem	cnblcld 14703*	Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
|- D = ( abs o. - )   =>    |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x e. CC | ( 0 D x ) <_ R } )
Theorem	unicntopcntop 14704	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
|- CC = U. ( MetOpen ` ( abs o. - ) )
Theorem	cnopncntop 14705	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
|- CC e. ( MetOpen ` ( abs o. - ) )
Theorem	reopnap 14706*	The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
|- ( A e. RR -> { w e. RR | w # A } e. ( topGen ` ran (,) ) )
Theorem	remetdval 14707	Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ( ( A e. RR /\ B e. RR ) -> ( A D B ) = ( abs ` ( A - B ) ) )
Theorem	remet 14708	The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- D e. ( Met ` RR )
Theorem	rexmet 14709	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- D e. ( *Met ` RR )
Theorem	bl2ioo 14710	A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ( ( A e. RR /\ B e. RR ) -> ( A ( ball ` D ) B ) = ( ( A - B ) (,) ( A + B ) ) )
Theorem	ioo2bl 14711	An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) = ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) )
Theorem	ioo2blex 14712	An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) e. ran ( ball ` D ) )
Theorem	blssioo 14713	The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ran ( ball ` D ) C_ ran (,)
Theorem	tgioo 14714	The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   &    |- J = ( MetOpen ` D )   =>    |- ( topGen ` ran (,) ) = J
Theorem	tgqioo 14715	The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
|- Q = ( topGen ` ( (,) " ( QQ X. QQ ) ) )   =>    |- ( topGen ` ran (,) ) = Q
Theorem	resubmet 14716	The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
|- R = ( topGen ` ran (,) )   &    |- J = ( MetOpen ` ( ( abs o. - ) |` ( A X. A ) ) )   =>    |- ( A C_ RR -> J = ( R ↾t A ) )
Theorem	tgioo2cntop 14717	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( topGen ` ran (,) ) = ( J ↾t RR )
Theorem	rerestcntop 14718	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- R = ( topGen ` ran (,) )   =>    |- ( A C_ RR -> ( J ↾t A ) = ( R ↾t A ) )
Theorem	addcncntoplem 14719*	Lemma for addcncntop 14720, subcncntop 14721, and mulcncntop 14722. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- .+ : ( CC X. CC ) --> CC   &    |- ( ( a e. RR+ /\ b e. CC /\ c e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - b ) ) < y /\ ( abs ` ( v - c ) ) < z ) -> ( abs ` ( ( u .+ v ) - ( b .+ c ) ) ) < a ) )   =>    |- .+ e. ( ( J tX J ) Cn J )
Theorem	addcncntop 14720	Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- + e. ( ( J tX J ) Cn J )
Theorem	subcncntop 14721	Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- - e. ( ( J tX J ) Cn J )
Theorem	mulcncntop 14722	Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- x. e. ( ( J tX J ) Cn J )
Theorem	divcnap 14723*	Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- K = ( J ↾t { x e. CC | x # 0 } )   =>    |- ( y e. CC , z e. { x e. CC | x # 0 } |-> ( y / z ) ) e. ( ( J tX K ) Cn J )
Theorem	fsumcncntop 14724*	A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for B normally contains free variables k and x to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn K ) )   =>    |- ( ph -> ( x e. X |-> sum_ k e. A B ) e. ( J Cn K ) )
9.2.7  Topological definitions using the reals
Syntax	ccncf 14725	Extend class notation to include the operation which returns a class of continuous complex functions.
class -cn->
Definition	df-cncf 14726*	Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
|- -cn-> = ( a e. ~P CC , b e. ~P CC |-> { f e. ( b ^m a ) | A. x e. a A. e e. RR+ E. d e. RR+ A. y e. a ( ( abs ` ( x - y ) ) < d -> ( abs ` ( ( f ` x ) - ( f ` y ) ) ) < e ) } )
Theorem	cncfval 14727*	The value of the continuous complex function operation is the set of continuous functions from A to B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
|- ( ( A C_ CC /\ B C_ CC ) -> ( A -cn-> B ) = { f e. ( B ^m A ) | A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( f ` x ) - ( f ` w ) ) ) < y ) } )
Theorem	elcncf 14728*	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
|- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) ) )
Theorem	elcncf2 14729*	Version of elcncf 14728 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
|- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) ) )
Theorem	cncfrss 14730	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( F e. ( A -cn-> B ) -> A C_ CC )
Theorem	cncfrss2 14731	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( F e. ( A -cn-> B ) -> B C_ CC )
Theorem	cncff 14732	A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- ( F e. ( A -cn-> B ) -> F : A --> B )
Theorem	cncfi 14733*	Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- ( ( F e. ( A -cn-> B ) /\ C e. A /\ R e. RR+ ) -> E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) )
Theorem	elcncf1di 14734*	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )   &    |- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )   =>    |- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )
Theorem	elcncf1ii 14735*	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)
|- F : A --> B   &    |- ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ )   &    |- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )   =>    |- ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )
Theorem	rescncf 14736	A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- ( C C_ A -> ( F e. ( A -cn-> B ) -> ( F |` C ) e. ( C -cn-> B ) ) )
Theorem	cncfcdm 14737	Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( C C_ CC /\ F e. ( A -cn-> B ) ) -> ( F e. ( A -cn-> C ) <-> F : A --> C ) )
Theorem	cncfss 14738	The set of continuous functions is expanded when the codomain is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
|- ( ( B C_ C /\ C C_ CC ) -> ( A -cn-> B ) C_ ( A -cn-> C ) )
Theorem	climcncf 14739	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. ( A -cn-> B ) )   &    |- ( ph -> G : Z --> A )   &    |- ( ph -> G ~~> D )   &    |- ( ph -> D e. A )   =>    |- ( ph -> ( F o. G ) ~~> ( F ` D ) )
Theorem	abscncf 14740	Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- abs e. ( CC -cn-> RR )
Theorem	recncf 14741	Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- Re e. ( CC -cn-> RR )
Theorem	imcncf 14742	Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- Im e. ( CC -cn-> RR )
Theorem	cjcncf 14743	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- * e. ( CC -cn-> CC )
Theorem	mulc1cncf 14744*	Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- F = ( x e. CC |-> ( A x. x ) )   =>    |- ( A e. CC -> F e. ( CC -cn-> CC ) )
Theorem	divccncfap 14745*	Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
|- F = ( x e. CC |-> ( x / A ) )   =>    |- ( ( A e. CC /\ A # 0 ) -> F e. ( CC -cn-> CC ) )
Theorem	cncfco 14746	The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- ( ph -> F e. ( A -cn-> B ) )   &    |- ( ph -> G e. ( B -cn-> C ) )   =>    |- ( ph -> ( G o. F ) e. ( A -cn-> C ) )
Theorem	cncfmet 14747	Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
|- C = ( ( abs o. - ) |` ( A X. A ) )   &    |- D = ( ( abs o. - ) |` ( B X. B ) )   &    |- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( A C_ CC /\ B C_ CC ) -> ( A -cn-> B ) = ( J Cn K ) )
Theorem	cncfcncntop 14748	Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- K = ( J ↾t A )   &    |- L = ( J ↾t B )   =>    |- ( ( A C_ CC /\ B C_ CC ) -> ( A -cn-> B ) = ( K Cn L ) )
Theorem	cncfcn1cntop 14749	Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( CC -cn-> CC ) = ( J Cn J )
Theorem	cncfmptc 14750*	A constant function is a continuous function on CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
|- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( x e. S |-> A ) e. ( S -cn-> T ) )
Theorem	cncfmptid 14751*	The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
|- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) e. ( S -cn-> T ) )
Theorem	cncfmpt1f 14752*	Composition of continuous functions. -cn-> analogue of cnmpt11f 14452. (Contributed by Mario Carneiro, 3-Sep-2014.)
|- ( ph -> F e. ( CC -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( X -cn-> CC ) )
Theorem	cncfmpt2fcntop 14753*	Composition of continuous functions. -cn-> analogue of cnmpt12f 14454. (Contributed by Mario Carneiro, 3-Sep-2014.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> F e. ( ( J tX J ) Cn J ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A F B ) ) e. ( X -cn-> CC ) )
Theorem	addccncf 14754*	Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- F = ( x e. CC |-> ( x + A ) )   =>    |- ( A e. CC -> F e. ( CC -cn-> CC ) )
Theorem	idcncf 14755	The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 14751 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
|- F = ( x e. CC |-> x )   =>    |- F e. ( CC -cn-> CC )
Theorem	sub1cncf 14756*	Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
|- F = ( x e. CC |-> ( x - A ) )   =>    |- ( A e. CC -> F e. ( CC -cn-> CC ) )
Theorem	sub2cncf 14757*	Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
|- F = ( x e. CC |-> ( A - x ) )   =>    |- ( A e. CC -> F e. ( CC -cn-> CC ) )
Theorem	cdivcncfap 14758*	Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
|- F = ( x e. { y e. CC | y # 0 } |-> ( A / x ) )   =>    |- ( A e. CC -> F e. ( { y e. CC | y # 0 } -cn-> CC ) )
Theorem	negcncf 14759*	The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- F = ( x e. A |-> -u x )   =>    |- ( A C_ CC -> F e. ( A -cn-> CC ) )
Theorem	negfcncf 14760*	The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- G = ( x e. A |-> -u ( F ` x ) )   =>    |- ( F e. ( A -cn-> CC ) -> G e. ( A -cn-> CC ) )
Theorem	mulcncflem 14761*	Lemma for mulcncf 14762. (Contributed by Jim Kingdon, 29-May-2023.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   &    |- ( ph -> V e. X )   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> F e. RR+ )   &    |- ( ph -> G e. RR+ )   &    |- ( ph -> S e. RR+ )   &    |- ( ph -> T e. RR+ )   &    |- ( ph -> A. u e. X ( ( abs ` ( u - V ) ) < S -> ( abs ` ( ( ( x e. X |-> A ) ` u ) - ( ( x e. X |-> A ) ` V ) ) ) < F ) )   &    |- ( ph -> A. u e. X ( ( abs ` ( u - V ) ) < T -> ( abs ` ( ( ( x e. X |-> B ) ` u ) - ( ( x e. X |-> B ) ` V ) ) ) < G ) )   &    |- ( ph -> A. u e. X ( ( ( abs ` ( [_ u / x ]_ A - [_ V / x ]_ A ) ) < F /\ ( abs ` ( [_ u / x ]_ B - [_ V / x ]_ B ) ) < G ) -> ( abs ` ( ( [_ u / x ]_ A x. [_ u / x ]_ B ) - ( [_ V / x ]_ A x. [_ V / x ]_ B ) ) ) < E ) )   =>    |- ( ph -> E. d e. RR+ A. u e. X ( ( abs ` ( u - V ) ) < d -> ( abs ` ( ( ( x e. X |-> ( A x. B ) ) ` u ) - ( ( x e. X |-> ( A x. B ) ) ` V ) ) ) < E ) )
Theorem	mulcncf 14762*	The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A x. B ) ) e. ( X -cn-> CC ) )
Theorem	expcncf 14763*	The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( CC -cn-> CC ) )
Theorem	cnrehmeocntop 14764*	The canonical bijection from ( RR X. RR ) to CC described in cnref1o 9716 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if ( RR X. RR ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
|- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) )   &    |- J = ( topGen ` ran (,) )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- F e. ( ( J tX J ) Homeo K )
Theorem	cnopnap 14765*	The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
|- ( A e. CC -> { w e. CC | w # A } e. ( MetOpen ` ( abs o. - ) ) )
PART 10  BASIC REAL AND COMPLEX ANALYSIS
10.1  Continuity
Theorem	addcncf 14766*	The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A + B ) ) e. ( X -cn-> CC ) )
Theorem	subcncf 14767*	The subtraction of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )
Theorem	divcncfap 14768*	The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> { y e. CC | y # 0 } ) )   =>    |- ( ph -> ( x e. X |-> ( A / B ) ) e. ( X -cn-> CC ) )
Theorem	maxcncf 14769*	The maximum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 18-Jul-2025.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )   =>    |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )
Theorem	mincncf 14770*	The minimum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 19-Jul-2025.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )   =>    |- ( ph -> ( x e. X |-> inf ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )
10.1.1  Dedekind cuts
Theorem	dedekindeulemuub 14771*	Lemma for dedekindeu 14777. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A e. U )   =>    |- ( ph -> A. z e. L z < A )
Theorem	dedekindeulemub 14772*	Lemma for dedekindeu 14777. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR A. y e. L y < x )
Theorem	dedekindeulemloc 14773*	Lemma for dedekindeu 14777. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> A. x e. RR A. y e. RR ( x < y -> ( E. z e. L x < z \/ A. z e. L z < y ) ) )
Theorem	dedekindeulemlub 14774*	Lemma for dedekindeu 14777. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR ( A. y e. L -. x < y /\ A. y e. RR ( y < x -> E. z e. L y < z ) ) )
Theorem	dedekindeulemlu 14775*	Lemma for dedekindeu 14777. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR ( A. q e. L q < x /\ A. r e. U x < r ) )
Theorem	dedekindeulemeu 14776*	Lemma for dedekindeu 14777. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( A. q e. L q < A /\ A. r e. U A < r ) )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A. q e. L q < B /\ A. r e. U B < r ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> F. )
Theorem	dedekindeu 14777*	A Dedekind cut identifies a unique real number. Similar to df-inp 7526 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E! x e. RR ( A. q e. L q < x /\ A. r e. U x < r ) )
Theorem	suplociccreex 14778*	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8092 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
|- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B < C )   &    |- ( ph -> A C_ ( B [,] C ) )   &    |- ( ph -> E. x x e. A )   &    |- ( ph -> A. x e. ( B [,] C ) A. y e. ( B [,] C ) ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) )   =>    |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
Theorem	suplociccex 14779*	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8092 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
|- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B < C )   &    |- ( ph -> A C_ ( B [,] C ) )   &    |- ( ph -> E. x x e. A )   &    |- ( ph -> A. x e. ( B [,] C ) A. y e. ( B [,] C ) ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) )   =>    |- ( ph -> E. x e. ( B [,] C ) ( A. y e. A -. x < y /\ A. y e. ( B [,] C ) ( y < x -> E. z e. A y < z ) ) )
Theorem	dedekindicclemuub 14780*	Lemma for dedekindicc 14787. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> C e. U )   =>    |- ( ph -> A. z e. L z < C )
Theorem	dedekindicclemub 14781*	Lemma for dedekindicc 14787. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )
Theorem	dedekindicclemloc 14782*	Lemma for dedekindicc 14787. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( E. z e. L x < z \/ A. z e. L z < y ) ) )
Theorem	dedekindicclemlub 14783*	Lemma for dedekindicc 14787. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E. x e. ( A [,] B ) ( A. y e. L -. x < y /\ A. y e. ( A [,] B ) ( y < x -> E. z e. L y < z ) ) )
Theorem	dedekindicclemlu 14784*	Lemma for dedekindicc 14787. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E. x e. ( A [,] B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
Theorem	dedekindicclemeu 14785*	Lemma for dedekindicc 14787. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   &    |- ( ph -> C e. ( A [,] B ) )   &    |- ( ph -> ( A. q e. L q < C /\ A. r e. U C < r ) )   &    |- ( ph -> D e. ( A [,] B ) )   &    |- ( ph -> ( A. q e. L q < D /\ A. r e. U D < r ) )   &    |- ( ph -> C < D )   =>    |- ( ph -> F. )
Theorem	dedekindicclemicc 14786*	Lemma for dedekindicc 14787. Same as dedekindicc 14787, except that we merely show x to be an element of ( A [,] B ). Later we will strengthen that to ( A (,) B ). (Contributed by Jim Kingdon, 5-Jan-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E! x e. ( A [,] B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
Theorem	dedekindicc 14787*	A Dedekind cut identifies a unique real number. Similar to df-inp 7526 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E! x e. ( A (,) B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
10.1.2  Intermediate value theorem
Theorem	ivthinclemlm 14788*	Lemma for ivthinc 14797. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E. q e. ( A [,] B ) q e. L )
Theorem	ivthinclemum 14789*	Lemma for ivthinc 14797. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E. r e. ( A [,] B ) r e. R )
Theorem	ivthinclemlopn 14790*	Lemma for ivthinc 14797. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   &    |- ( ph -> Q e. L )   =>    |- ( ph -> E. r e. L Q < r )
Theorem	ivthinclemlr 14791*	Lemma for ivthinc 14797. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
Theorem	ivthinclemuopn 14792*	Lemma for ivthinc 14797. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   &    |- ( ph -> S e. R )   =>    |- ( ph -> E. q e. R q < S )
Theorem	ivthinclemur 14793*	Lemma for ivthinc 14797. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. r e. ( A [,] B ) ( r e. R <-> E. q e. R q < r ) )
Theorem	ivthinclemdisj 14794*	Lemma for ivthinc 14797. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> ( L i^i R ) = (/) )
Theorem	ivthinclemloc 14795*	Lemma for ivthinc 14797. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. R ) ) )
Theorem	ivthinclemex 14796*	Lemma for ivthinc 14797. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E! z e. ( A (,) B ) ( A. q e. L q < z /\ A. r e. R z < r ) )
Theorem	ivthinc 14797*	The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
Theorem	ivthdec 14798*	The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` B ) < U /\ U < ( F ` A ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` y ) < ( F ` x ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
Theorem	ivthreinc 14799*	Restating the intermediate value theorem. Given a hypothesis stating the intermediate value theorem (in a strong form which is not provable given our axioms alone), provide a conclusion similar to the theorem as stated in the Metamath Proof Explorer (which is also similar to how we state the theorem for a strictly monotonic function at ivthinc 14797). Being able to have a hypothesis stating the intermediate value theorem will be helpful when it comes time to show that it implies a constructive taboo. This version of the theorem requires that the function F is continuous on the entire real line, not just ( A [,] B ) which may be an unnecessary condition but which is sufficient for the way we want to use it. (Contributed by Jim Kingdon, 7-Jul-2025.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> F e. ( RR -cn-> RR ) )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
Theorem	hovercncf 14800	The hover function is continuous. By hover function, we mean a a function which starts out as a line of slope one, is constant at zero from zero to one, and then resumes as a slope of one. (Contributed by Jim Kingdon, 20-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- F e. ( RR -cn-> RR )